Algebra Examples

$\frac{4 y + 3}{- 2 y + 5} - \frac{7}{9 y^{4}}$

Step 1

To write $\frac{4 y + 3}{- 2 y + 5}$ as a fraction with a common denominator, multiply by $\frac{9 y^{4}}{9 y^{4}}$ .

$\frac{4 y + 3}{- 2 y + 5} \cdot \frac{9 y^{4}}{9 y^{4}} - \frac{7}{9 y^{4}}$

Step 2

To write $- \frac{7}{9 y^{4}}$ as a fraction with a common denominator, multiply by $\frac{- 2 y + 5}{- 2 y + 5}$ .

$\frac{4 y + 3}{- 2 y + 5} \cdot \frac{9 y^{4}}{9 y^{4}} - \frac{7}{9 y^{4}} \cdot \frac{- 2 y + 5}{- 2 y + 5}$

Step 3

Write each expression with a common denominator of $(- 2 y + 5) \cdot 9 y^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4 y + 3}{- 2 y + 5}$ by $\frac{9 y^{4}}{9 y^{4}}$ .

$\frac{(4 y + 3) (9 y^{4})}{(- 2 y + 5) (9 y^{4})} - \frac{7}{9 y^{4}} \cdot \frac{- 2 y + 5}{- 2 y + 5}$

Step 3.2

Multiply $\frac{7}{9 y^{4}}$ by $\frac{- 2 y + 5}{- 2 y + 5}$ .

$\frac{(4 y + 3) (9 y^{4})}{(- 2 y + 5) (9 y^{4})} - \frac{7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 3.3

Reorder the factors of $(- 2 y + 5) (9 y^{4})$ .

$\frac{(4 y + 3) (9 y^{4})}{9 y^{4} (- 2 y + 5)} - \frac{7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(4 y + 3) (9 y^{4}) - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(4 y \cdot 9 + 3 \cdot 9) y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.2

Multiply $9$ by $4$ .

$\frac{(36 y + 3 \cdot 9) y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.3

Multiply $3$ by $9$ .

$\frac{(36 y + 27) y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.4

Apply the distributive property.

$\frac{36 y \cdot y^{4} + 27 y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.5

Multiply $y$ by $y^{4}$ by adding the exponents.

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Step 5.5.1

Move $y^{4}$ .

$\frac{36 (y^{4} y) + 27 y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.5.2

Multiply $y^{4}$ by $y$ .

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Step 5.5.2.1

Raise $y$ to the power of $1$ .

$\frac{36 (y^{4} y^{1}) + 27 y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{36 y^{4 + 1} + 27 y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.5.3

Add $4$ and $1$ .

$\frac{36 y^{5} + 27 y^{4} - 7 (- 2 y + 5)}{9 y^{4} (- 2 y + 5)}$

Step 5.6

Apply the distributive property.

$\frac{36 y^{5} + 27 y^{4} - 7 (- 2 y) - 7 \cdot 5}{9 y^{4} (- 2 y + 5)}$

Step 5.7

Multiply $- 2$ by $- 7$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 7 \cdot 5}{9 y^{4} (- 2 y + 5)}$

Step 5.8

Multiply $- 7$ by $5$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (- 2 y + 5)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 y$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (- (2 y) + 5)}$

Step 6.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (- (2 y) - 1 (- 5))}$

Step 6.3

Factor $- 1$ out of $- (2 y) - 1 (- 5)$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (- (2 y - 5))}$

Step 6.4

Rewrite negatives.

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Step 6.4.1

Rewrite $- (2 y - 5)$ as $- 1 (2 y - 5)$ .

$\frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (- 1 (2 y - 5))}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{36 y^{5} + 27 y^{4} + 14 y - 35}{(9 y^{4}) (2 y - 5)}$

Step 6.4.3

Reorder factors in $- \frac{36 y^{5} + 27 y^{4} + 14 y - 35}{(9 y^{4}) (2 y - 5)}$ .

$- \frac{36 y^{5} + 27 y^{4} + 14 y - 35}{9 y^{4} (2 y - 5)}$